I've seen the other way for this question (Entire functions $f$ for which there exists a positive constant $M$ such that $|f(z)|\le M|\cos z|$), I just want to make sure that the same argument can be applied for the following question.
Find all entire functions $f(z)$ for which there exists a positive constant $M$ such that $$|f(z)|\geq M.|\sin z|\:\:\text{for all}\:\:z\in\mathbb{C}.$$
From the above assuption, I can say the zeros of $f(z)$ is a subset of zeros of $\sin z $. Define $g(z)=\frac{\sin z}{f(z)}$. Hence $|g(z)|\leq M$, so the isoalted singularities of $g(z)$, becomes removable and so $g(z)$ can be extended to an entire function which by Liouville's theorem it becomes constant.
Thus $$f(z)= k \sin z,\:\:\text{ for some nonzero constant }k.$$